A torsor (or principal homogeneous space) is, informally speaking, a mathematical structure quite similar to a group, but without a natural identity element. More formally, if is a group, a -torsor is a set on which acts simply and transitively, i.e., for every , there is a unique such that . Torsors are ubiquitous in mathematics, see for example this blog post by John Baez.

I noticed that one can define the notion of torsor without ever mentioning groups, and then recover the notion of a group from this, rather than vice-versa. I’ve never seen this formalized before, so I thought I’d take the time to write it down here. Please let me know if you’re aware of a reference! (Note added 9/18/23: Ben Steinberg points out that this is closely related to the notion of a heap — see below for additional details.)

I will coin a new term — proportion space — for the concept I’m about to describe. But we’ll see that a proportion space has an associated group which acts simply and transitively on , making into a -torsor, and conversely every torsor for a group is naturally a proportion space (with associated group isomorphic to ). So the concepts of torsor and proportion space are essentially equivalent, except for the fact that simply transitive group actions which differ by an automorphism of correspond to the same proportion structure on .